Superior highly composite number

In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to some positive power of the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.

The first 10 superior highly composite numbers and their factorization are listed.

Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.

For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have

Note that the product need not be computed indefinitely, because if p>2x{\displaystyle p>2^{x}} then ep(x)=0{\displaystyle e_{p}(x)=0}, so the product to calculate s(x){\displaystyle s(x)} can be terminated once p≥2x{\displaystyle p\geq 2^{x}}.

Also note that in the definition of ep(x){\displaystyle e_{p}(x)}, 1/x{\displaystyle 1/x} is analogous to ε{\displaystyle \varepsilon } in the implicit definition of a superior highly composite number.

This representation implies that there exist an infinite sequence of π1,π2,…∈P{\displaystyle \pi _{1},\pi _{2},\ldots \in \mathbb {P} } such that for the n-th superior highly composite number sn{\displaystyle s_{n}} holds

sn=∏i=1nπi{\displaystyle s_{n}=\prod _{i=1}^{n}\pi _{i}}

The first πi{\displaystyle \pi _{i}} are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in the OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.